Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

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trace to be equal to the total particle number N:

      (24)

      The average value of the interaction energy can then be written as:

      (25)

3-c. Time derivative

      As for the time derivative term, the function it contains can be written as:

      (26)

      In this summation, all terms involving the individual states j other than the state i (which is undergoing the derivation) lead to an expression of the type:

      (27)

      which equals 1 since this expression is the square of the norm of the state , which is simply the Fock state |nj = 1〉. As for the state i, it leads to a factor written in the form of a scalar product in the one-particle state space:

      (28)

3-d. Functional value

      Regrouping all these results, we can write the value of the functional S in the form:

(29)

4. Equations of motion

      We now vary the ket |θk(t)〉 according to:

      (30)

As in complement E_XV, we will only consider variations |δθk(t)〉 that lead to an actual variation of the ket ; those where |δθk(t)〉 is proportional to one of the occupied states |θl(t)〉 with l ≤ N yield no change for (or at the most to a phase change) and are thus irrelevant for the value of S. As we did in relations (32) or (69) of Complement E_XV, we assume that:

(31)

      where δf(t) is an infinitesimal time-dependent function.

The computation is then almost identical to that of § 2-b in Complement E_XV. When |θk(t)〉 varies according to (31), all the other occupied states remaining constant, the only changes in the first line of (29) come from the terms i = k. In the second line, the changes come from either the i = k terms, or the j = k terms. As the W₂(1,2) operator is symmetric with respect to the two particles, these variations are the same and their sum cancels the 1/2 factor. All these variations involve terms containing either the ket eiχ |δθk(t)〉, or the bra 〈δθk(t)|e–iχ. Now their sum must be zero for any value of χ, and this is only possible if each of the terms is zero. Inserting the variation (31) of |θk(t)〉, and canceling the term in e–iχ leads to the following equality:

(32)

As we recognize in the function to be integrated the Hartree-Fock potential operator WHF(1, t) defined in (21), we can write:

(33)

      with l > N.

4-a. Time-dependent Hartree-Fock equations

As the choice of the function δf(t) is arbitrary, for expression (33) to be zero for any δf(t) requires the function inside the curly brackets to be zero at all times t. Stationarity therefore requires the ket:

      (34)

      to have no components on any of the non-occupied states |θl(t)〉 with (l > N). In other words, stationarity will be obtained if, for all values of k & between 1 and N, we have:

      (35)

      where |ξk(t)〉 is any linear combination of the occupied states |θl(t)〉 (l < N). As we pointed out at the beginning of § 4, adding to one of the |θk(t)〉 a component on the already occupied individual states has no effect on the Af-particle state (aside from an eventual change of phase), and therefore does not change the value of S; consequently, the stationarity of this functional does not depend on the value of the ket |ξk(t)〉, which can be any ket, for example the zero ket.

      Finally, if the |θn(t)〉 are equal to the solutions |φn(t)〉 of the N equations:

(36)

the functional S is indeed stationary for all times. Furthermore, as we saw in Complement E_xv that WHF(t) is Hermitian, so is the operator on the right-hand side of (36). Consequently, the N kets |φn(t)〉 follow an evolution similar to the usual Schrödinger evolution, described by a unitary evolution operator (Complement F_III). Such an operator does not change either the norm nor the scalar products of the kets: if the kets |φn(t)〉x

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